Title: Settling the complexity of k-disjointness and the k-Hamming distance problems
Advisor: Shayan Oveis Gharan
Supervisory Committee: Shayan Oveis Gharan (Chair), Ioana Dumitriu (GSR, Math), Thomas Rothvoss, Anup Rao, and Paul Beame
Abstract: Suppose two parties, traditionally called Alice and Bob, are given respectively the inputs and to a function and are required to compute . Since each party only has one part of the input, they can compute only if some communication takes place between them. The communication complexity of a given function is the minimum amount of communication (in bits) needed to evaluate it on any input with high probability.
We study the communication complexity of two related problems, the -Hamming distance and -disjointness and give tight bounds to both of these problems: The -round communication complexity of the -disjointness problem is , whereas a tight bound holds for the -Hamming distance problem for any number of rounds.
The lower bound direction of our first result is obtained by proving a super-sum result on computing the OR of equality problems, which is the first of its kind. Using our second bound, we settle the complexity of various property testing problems such as -linearity, which was open since 2002 or earlier. Our lower bounds are obtained via information theoretic arguments and along the way we resolve a question conjectured by Erdős and Simonovits in 1982, which incidentally was studied even earlier by Blakley and Dixon in 1966.